Archive of posts filed under the Teaching category.

## Regents Recap — June, 2017: When Side-Side-Angle is Enough

Here is yet another mathematically erroneous question from New York’s June 2017 Geometry Regents exam.

At first this question seems straightforward.  There are several ways to determine if two triangles are similar, and the answer choices cover three of the basics: in (1) segment AB is parallel to segment ED, so congruent alternate interior angles can be used to show that the triangles are similar by Angle-Angle (AA); in (3) Side-Angle-Side (SAS) similarity can be used; and in (4), Side-Side-Side (SSS) similarity applies since all three pairs of sides are in proportion.

Presumably (2) is the answer choice that does not guarantee the triangles will be be similar, and according to the official scoring guide provided by the state (2) is the correct answer.  But as it turns out, (2) is also sufficient to guarantee that the triangles are similar.  This means that this question has no correct answer.

In (2), we have two pairs of sides in proportion and one pair of congruent angles (the vertical angles ECD and ACB).  This is the Side-Side-Angle (SSA) scenario, and because this set of information does not determine a unique triangle, SSA alone is not sufficient to establish that a pair of triangles are similar (or congruent).

But there is additional information to work with in this question.  The lengths of the sides of the triangles guarantee that angles B and D are both acute.  This is because there can be at most one non-acute angle in any triangle, which is necessarily the triangle’s largest angle, and the largest angle in a triangle must be opposite the triangle’s longest side.  Since angles B and D are not opposite their respective triangle’s longest side, they must be acute angles.  And it turns out that this additional piece of information allows us to conclude that the triangles are similar.

Here’s why.  Suppose you know the lengths of segments XY and YZ and the measure of an acute angle Z.  Depending on the length of XY, there are 0, 1, or 2 possible triangles XYZ.  Here’s a geometric representation of all the possibilities:

This explains why SSA fails to uniquely determine a triangle:  there may exist two different triangles consistent with the given information.

But if two triangles XYZ are possible, one of the triangles will have an obtuse angle at X and the other will have an acute angle at X.   This means that if we happen to know that angle X is acute, then only one triangle XYZ is possible, and so this set of information (SSA and the nature of the angles opposite the given sides) uniquely identifies a triangle and can be used to establish similarity (or congruence) among a pair of triangles.  Thus, the information in (2) is sufficient to conclude the triangle are similar, and so there is no correct answer to the above exam question.

Alternately, a more algebraic argument uses the Law of Sines.  From triangle ABC we get

$\frac{sinB}{7.2} = \frac{sinACB}{8.1}$

$sinB = \frac{8}{9} sinACB$

and from triangle EDC we get

$\frac{sinD}{2.4} = \frac{sinECD}{2.7}$

$sinD = \frac{8}{9} sinECD$

And since

$\angle{ACB} \cong \angle{ECD}$

we can conclude that

$sinB = sinD$

Generally speaking we can’t conclude that the measure of angle B is equal to the measure of angle D:  two angles with the same sine could be supplements or differ by a full revolution.  But since we know both angles are acute, we can conclude that

$m\angle{B} = m\angle{D}$

Thus, the triangles are similar by AA.  (This argument also shows that SSA together with knowledge of the nature of the angles is a congruence theorem.)

So, this high-stakes exam question has no correct answer.  And despite the Change.org petition started by a 16-year old student that made national news, the New York State Education Department refuses to issue a correction.  In fact, they refuse to acknowledge the indisputable fact that this question has no correct answer, perhaps because they don’t want to admit that a third question on this exam (see question 14 and question 22) has been determined to be mathematically erroneous.

UPDATE:  All the media attention apparently convinced the NYSED to award full credit to all test takers for this erroneous question.  Due to the discrepancy in wording, of course (link).

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## Regents Recap — June, 2017: Trouble with Dilations (and Logic)

The emphasis on transformations in Common Core Geometry has proven to be a challenge for the creators of the New York State Regents.  Here’s the latest example.

This is a tricky question.  So tricky, in fact, that it tripped up those responsible for creating this exam.

Dilation is a similarity mapping (assuming, as we do, that the scale factor is non-zero), and translation is a congruence mapping.  Thus, any composition of the two will be a similarity mapping, but not necessarily a congruence mapping.  So in the above question, statement II will always be true, and statements I and IV are not always true.

Statement III requires closer attention.  Under most circumstances, translations and dilations map lines to parallel lines, and so the same would be true of their compositions.  However, if the center of dilation lies on a given line, or the translation is parallel to the given line, then that line will be mapped onto itself under the transformation.

This means that the answer to this test question hinges on the question, “Is a line parallel to itself?”

If the answer is yes, then statement III will always be true, and so (3) II and III will be the correct answer.  If the answer is no, then statement III won’t always be true. and so (1) II only will be the correct answer.

So which is the correct answer?  Well, that’s tricky, too.  The answer key provided by New York state originally gave (3) as the correct answer.  But several days later, the NYS Department of Education issued a memo instructing graders to accept both (1) and (3) as correct.  Apparently, the state isn’t prepared to take a stance on this issue.

Their final decision is amusing, as these two answer choices are mutually exclusive:  either statement III is always true or it isn’t always true.  It can’t be both.  Those responsible for this exam are trying to get away with quietly asserting that (P and not P) can be true!

Oddly enough, this wasn’t the only place on this very exam where this issue arose.  Here’s question 6:Notice that this question directly acknowledges that the location of the center of dilation impacts whether or not a line is mapped to a parallel line.  It’s not entirely correct (a center’s location on the line, not the segment, is what matters) but it demonstrates some of the knowledge that was lacking in question 14.  How, then, did the problem with question 14 slip through?

As is typical, the state provided a meaningless and generic explanation for the error:  this problem was a result of discrepancies in wording.  But there are no discrepancies in wording here.  This is simply a careless error, one that should have been caught early in the test production process, and one that would have been caught if production of these exams were taken more seriously.

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## Regents Recap — June, 2017: More Mathematical Misunderstanding

Far too often questions on New York State math exams demonstrate a disturbing lack of content knowledge.

Consider this multiple choice item from the June, 2017 Common Core Geometry exam.

Instead of testing a student’s understanding of triangle congruence, this question exposes a serious lack of mathematical understanding among the exam creators.

A superficial reading of the problem suggests that (3) is the correct answer.  In (1), the two triangles share only three pairs of congruent angles; in (2), two sides and a non-included angle are congruent in each triangle; and in (4), the triangles share only one pair of congruent sides and one pair of congruent angles.  None of these scenarios (AAA, SSA, SA) seems sufficient to guarantee that the triangles are congruent.  And in (3), the triangles have one pair of congruent sides and two pairs of congruent angles; this (ASA or SAA) is sufficient to conclude the triangles are congruent, so (3) is apparently the correct answer.

But closer inspection shows that, in fact, (1) and (2) are correct as well.

Consider choice (1).  While it’s not exactly clear what it means to map angle A onto angle D, it must require that point A gets mapped to point D.  Similarly, point C must be mapped to point F.  If a rigid motion maps to D and C to F, then segment AC must be congruent to segment DF.  We now have one pair of congruent sides and three pairs of congruent angles: the triangles are congruent (by ASA or SAA), and choice (1) is a correct answer.

In (2), we are given that segment AC is mapped onto segment DF.  This means that point A gets mapped to point D and point C gets mapped to point F.  And since segment BC is mapped onto segment EF, we know that B is mapped onto E.  Therefore, the vertices of triangle ABC are mapped via rigid motion onto the vertices of triangle DEF.  This is sufficient to conclude that the triangles are congruent, and choice (2) is also a correct answer. (It’s also worth noting that, since the triangles are given as acute, SSA is actually sufficient to guarantee that the triangles are congruent.  This mathematical error turned up in a separate question on this exam.)

As it stands, the only option that is not a correct answer to this question is (4).

Within a few days, the NYS Education Department issued a directive to count all answers to this question as correct.  As is typical, no admission of an error was made:  the problem was blamed on discrepancies in wording.  Of course, there are no discrepancies in wording here:  this problem as written, reviewed, edited, and ultimately published is simply mathematically incorrect.  Its existence demonstrates a fundamental misunderstanding of the underlying concepts.

This isn’t the first time an erroneous question has made it onto one of these high-stakes Regents exams.  In fact, there were at least three mathematically invalid questions on this exam alone!  Over the past five years I’ve documented many others, and each time it happens, it raises serious questions:  Questions about the validity of these exams, how they are experienced by students, how they are scored, and the lack of accountability for those in charge.

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## MfA Summer Think 2017

I’m excited to be participating in the inaugural Summer Think, a teacher-led summer conference hosted by Math for America.

The theme of the conference is Big Ideas In and Out of the Classroom.   I’ll be delivering the opening keynote, Connecting Big Ideas In and Out of the Classroom, and will also be running a related workshop later in the conference.  I’m honored to be providing the opening remarks for this event, which is undoubtedly the first of many.

The Summer Think has been organized and orchestrated by teachers from Math for America’s various fellowship programs, and features two dozen sessions proposed and facilitated by teachers.  In addition to the underlying support offered through their fellowship programs, MfA has provided organizational support for the conference.

Teacher-led, teacher-driven professional development lies at the heart of Math for America’s programs, and the fact that over 50 teachers are participating in a conference less than two weeks after the end of the school year speaks to the impact it has on MfA’s teachers.